Ecology

Fin and sperm whales residing or circulating in the Mediterranean Sea are exposed to biological and chemical hazard due to the increasing anthropogenic impact. In particular, most of the coastal areas bordering with the Sanctuary is heavily populated and full of commercial, touristic and military ports and industrial areas. As a consequence, a range of diverse human activities exerts several actual and potential threats to cetacean populations in the Sanctuary, including habitat degradation, urban, tourist, industrial, and agricultural development, intense maritime traffic, military exercises and oil and gas exploration, just to mention the most important ones.
This study provides background information on the occurrence and concentration of parasites and bacterial infections/communities as well a first investigation of heavy metals and organic pollutants in faecal samples from fin and sperm whale Mediterranean subpopulations within the Pelagos Sanctuary.
Here, a modified MINI-FLOTAC technique in combination with FILL-FLOTAC were used for parasitological detection of the cysts in the faecal samples of fin and sperm whales. Although this technique has never been used before for whale faecal samples, it has successfully been used in previous coprological surveys for the detection of gastrointestinal parasites in other marine animals as the loggerhead sea turtles (Caretta caretta)23,24. The MINI-FLOTAC can be considered as one of the most accurate methods for coprological diagnosis of endoparasite infections and cysts/eggs counting nowadays available in veterinary medicine25. It allowed an accurate and reliable detection of Blastocystis cysts in both fin and sperm faecal samples. Molecular analysis, sequencing and phylogenetic analysis confirmed the obtained results.
Blastocystis is a common intestinal protozoan parasite reported in several animals, e.g., humans, livestock, dogs, amphibians, reptiles, birds and even insects26,27,28. Although it possesses pathogenic potential, its virulence mechanisms in humans are still not well understood29. Blastocystis seems to be linked to Irritable Bowel Syndrome, i.e., a functional disorder mainly consisting in chronic or recurrent abdominal pain due to altered intestinal habits30. Studying the small subunit ribosomal RNA (SSU-rDNA) gene, several authors identified at least 22 different Blastocystis subtypes (ST) in a variety of animals, humans included, i.e., from ST1 to ST17, ST21, and ST23 to ST26 (Ref.26). To date, human Blastocystis isolates are classified into 10 ST (i.e., ST1-ST9 and ST12) that, with the only exception of ST9, have been identified also in other animals31. According to Parkar et al.32, Blastocystis has the potential to spread through human-to-human, animal-to-human, and human-to-animal contact.
Few similar parasitological investigations have been conducted in the past and are currently available in the literature. Hermosilla et al.33 detected three protozoan parasites (i.e., Giardia sp., Balantidium sp., Entamoeba sp.) and helminth parasites in individual faecal samples from wild fin (n. 10), sperm (n. 4), blue (Balaenoptera musculus; n. 2) and sei (Balaenoptera borealis; n. 1) Atlantic whale subpopulations from the Azores Islands, Portugal. Protozoan parasites (Giardia sp., Balantidium sp., Cistoisospora-like indet.) and helminth parasites were also found in individual faecal samples of wild sperm whales inhabiting Mediterranean Sea waters surrounding the Balearic Archipelago, Spain34. Out of these, three of herein detected parasites clearly bear anthropozoonotic potential, i.e., Anisakis, Balantidium and Giardia34.
In the present work, Blastocystis has been found in fin and sperm whale samples and, to the best of our knowledge, this is the first time that this protozoan genus is reported for any cetacean species. Therefore, this finding represents the first new host record for fin and sperm whales. Blastocystis ST3 was the only subtype found in fin and sperm whales. Molecular studies in human samples showed the occurrence of ST1–ST9, with ST3 as the most prevalent subtype35,36. Indeed, ST3 is the Blastocystis subtype with the highest prevalence in humans worldwide and probably represents the human species-specific ST (Ref.37). Consequently, animals harbouring ST3 may thus mirror environmental contamination by humans, confirming the zoonotic potential of animals for Blastocystis human infections. Unlike33,34, no eggs of helminths were found in our faecal samples.
Variations in parasites composition and prevalence might be related to several factors such as dietary differences, the parasite life cycle, the availability of hosts necessary to complete their life cycle, the interactions between parasite species, the host immune response, and the host population density23. Moreover, parasites can also spread in different way in animal populations in the wild, particularly when they act together with ecological, biological, and anthropogenic factors38.
The occurrence in whales of parasites with a zoonotic potential like Giardia or Balantidium, most probably due to coastal waters contaminated by sewage, agricultural and urban run-off, has been already reported elsewhere39,40,41,42,43. Furthermore, human excretions from increasing number of pleasure boats, fishing and whale watching boats could be an additional form of contamination. Finally, the intense maritime traffic in the Mediterranean Sea, the percentage of which is higher than in other oceans44, represents another source of contamination. In all cases, results highlight that human activities play an important role for the widespread of these pathogens.
No bacterial pathogen of human or terrestrial animal origin has been detected both by targeted PCR and by Illumina high throughput sequencing. This difference could be due to the lower survival rate of bacteria in the sea environment, compared to protozoan parasites45.
Previous works reported the occurrence of human pathogens in stranded common minke whale (Balaenoptera acutorostrata) from Philippines46 and killer whale respiratory microbiome in North Pacific47. Although the relatively low number of samples cannot exclude potential risk of transmission of human and zoonotic pathogenic bacteria to cetaceans in the surveyed area, our results suggest to focus on microbiological analyses to track potential internal waterborne pathogens to the ones able to form cysts (like parasites) or other forms of resistance (like spore-forming bacteria) that are more likely to survive for longer period in the seawater.
The dominance of Bacterioidetes and Firmicutes (common with other terrestrial mammals), the baleen-specific higher number of Spirochaetes and the lower of Proteobacteria characterized both species, as also reported elsewhere21. Moreover, differences of some taxa related with the diverse diet were confirmed: in the case of sperm whale, whose nutrition is based on cephalopods, a higher proportion of Synergistetes was observed in faecal samples, whereas faecal samples from fin whales had a higher level of Spirochaetes compared those from sperm whales. These findings are in agreement with Erwin et al.13. The Synergistetes phylum includes gram negative, anaerobic, rod-shaped bacteria, widely distributed in terrestrial and aquatic environments, including host-associated with mammals48. Within this phylum, Synergistaceae family and Pyramidobacter genus OTUs were particularly dominant among sperm whale microbiome (Fig. 4). However, no correlation with potential pathogenicity could be drawn from the presence of these specific OTUs, considering their ubiquity in oral and gut mucosa of marine and terrestrial animals, despite some of the genus belonging to Synergistaceae family (e.g. Cloacibacillus spp.) are considered opportunistic pathogens49. About potential health implication of Spirochaetes, similar conclusion than for Synergistetes could be drawn: Treponema sp., found as dominant genus in fin whales, were found in healthy baleen whales by Sanders et al.21 so as among more dynamical OTU in stranded right whales50Sphaerochaeta spp. associated with healthy cetacean monitored oral cavity microbiome14. Moreover, fin whale faeces also showed a higher proportion of taxa that are also enriched in terrestrial herbivores, like Lentisphaere, Verrucomicrobia, Actinobacteria and Tenericutes, as also reported elsewhere21. Although differences in species sampled and habitats compared to previous studies, we found confirmation of both species and diet-influenced gut microbiota composition. Notably Akkermansia (one of the dominant Verrucomicrobia OTUs) and Coriobacteriaceae (dominant family among Actinobacteria phylum) includes typical holobiont of terrestrial and marine mammals, but also some pathobiont, so far confirmed only for humans51. Due to the wide distribution of some of Synergistetes and Spirochaetes phyla, it is not possible to establish if their presence could be ascribed exclusively to an anthropogenic impact; however, it is worth of interest that some of the genus found in both whale species and belonging to these phyla include opportunistic pathogens whose virulence for marine mammals still need to be confirmed. Interestingly, some archaeal sequences related to the Thermoplasmatales order were also found. This confirms what already reported by Sanders et al.21, i.e., that archaea belonging to this order may have a role as methane producer from methylated amines in baleen gut, differently from methanogenic archaea belonging to other orders that typically colonize the gut of terrestrial mammals, including humans.
Therefore, the two sampled species harboured typical gut microbiome belonging to fin whale and sperm whale groups. These data extend the spectrum of surveyed whales gut microbiome to previously unsampled species and confirms that NGS analyses could be a useful tool to retrieve information on the health status of wild whales.
While the concentration of 16 U.S. EPA priority PAHs and of 29 PCBs, being always  More

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Spatial map of Chinese protected areas
The database of protected areas (PA) distribution in China and a digitized spatial map thereof were compiled from Zhao et al.40 and Zhang et al.41. In total, we obtained the information of 2622 protected areas in China, which also included marine reserves. In order to evaluate the representation of terrestrial protected areas, we excluded marine reserves from our analyses. We also excluded Taiwan because we did not have the spatial distribution data for nature reserves in Taiwan. Finally, we had the boundary information of 2572 protected areas covering about 15.2% land area in China.
Species’ range maps
Range maps of threatened vertebrates (birds, mammals, amphibians and reptiles) were obtained from the IUCN’s Red List42. Distribution data of threatened plants were compiled from Flora of China, Atlas of woody plants in China, provincial and local floras, checklists of nature reserves, various inventory reports across China and peer-reviewed papers. We obtained the information for critically endangered (CR), endangered (EN) and vulnerable (VU) species. The conservation status of vertebrates was obtained from IUCN Red List42, while that of plants was obtained from Qin et al.43. In total, we obtained the distribution information of 103 birds, 86 mammals, 134 amphibians, 50 reptiles, and 2983 plants in China (see Supplementary Data 1). We estimated the number of species in each PA by overlaying the map of PA with the species’ range maps in ArcGIS 10.2 (ESRI, Redlands, CA). In order to validate the distribution of species, we further verified the presence of species in respective PA by checking their inventory reports.
Human footprint data
In order to measure the extent of human pressure on the protected areas, we obtained the most comprehensive global map of human pressure i.e., human footprint (HFP) from https://wcshumanfootprint.org. The human footprint measures the cumulative impact of direct pressures on environment from human activities and is based on data from built environments, agricultural lands, pasture lands, human population density, night-time lights, railways, roads and navigable waterways44. It is one of the most complete and finest terrestrial datasets on cumulative human pressure on the environment. The human footprint maps of two time periods (1993 and 2009) are available at present. We downloaded the maps of both time periods at the spatial resolution of 1 km × 1 km to quantify the change in human pressure within Chinese protected areas over a 16-year period. It should, however, be noted that any point estimate of the change in HFP might include errors due to the resolution and reliability of the component layers. For example, one of the components of HFP is the night-time lights, which changed over time from incandescent to mercury vapor to light emitting diode. This means that the change in night light is due to more than development. As a result, the systemic bias in regional economy could likely cause low HFP in wealthy as compared to rural areas. While this issue does not invalidate our analyses, such comparisons should be applied with caution.
Climate data
In order to represent the climatic conditions of the past and the present, we obtained 20 years climate data comprising monthly minimum temperature, monthly maximum temperature and monthly precipitation for two time periods (past: 1961–1970 and present: 2010–2019). We used 10 years window for each time period to capture the variability in climatic conditions in order to prevent over- or underestimation of the past and present climate. In total, we obtained 12 months × 10 years × 3 variables × 2 time periods = 720 global raster layers from the Climate Research Unit (CRU TS v. 4.04) database (http://www.cru.uea.ac.uk/data) at the spatial resolution of 0.5° × 0.5°45. We then calculated the monthly mean values for the three variables for each time period separately. From these monthly mean values, we estimated mean annual temperature (MAT), mean temperature of warmest quarter (MTWQ), mean temperature of coldest quarter (MTCQ), mean annual precipitation (MAP), precipitation of driest quarter (PDQ) and precipitation of wettest quarter (PWQ) for each time period using biovars function in the R package ‘dismo’46. We then estimated the average value of climate variables in each PA using zonal.stats function in the R package ‘spatialEco’47. In order to reduce dimensionality and collinearity of the 6 climate variables, we performed principal component analysis (PCA) using prcomp function in R v4.0.248. Following Carroll et al.37, we used climate data for both past and present based on the first 2 PCA axes, which explained 89.4–89.8 % of the variance (Supplementary Tables 1-2). Additionally, we also estimated the change in mean annual temperature to identify PAs that have experienced climate warming higher than the Paris Agreement threshold of 1.5 °C. All the analyses were performed in R version 4.0.248.
Vulnerability mapping
We calculated three indices of vulnerability within each protected area: (i) species vulnerability, (ii) anthropogenic vulnerability, and (iii) climate vulnerability. In order to measure species vulnerability, we first assigned numerical value to each IUCN threat category using a geometric progression49,50. We gave scores of 2, 4, and 8 to species belonging to categories VU, EN, and CR, respectively. We, then, summed the score of all species in each PA and standardized the value to the range of 0–1 using minimum–maximum normalization. We performed these steps separately for birds, mammals, amphibians, reptiles and plants to calculate the vulnerability score of each group. We also calculated the cumulative score by combining the total scores of all five groups. Values close to 0 indicated low species vulnerability and values close to 1 indicated high species vulnerability. Although species diversity is highly correlated with the species vulnerability metric used herein (Pearson’s correlation coefficient = 0.94 − 0.99, p  More

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Overview
In earlier work34, epidemiological models are broadly divided into two large categories, called forecasting and mechanistic. The former models fit a specific curve to the data and then attempt to predict the dynamics of the quantity under consideration. The most well known mechanistic models are the SIR-type models. As noted by Holmadahl and Buckee34, the mechanistic models involve substantially more complicated mathematical machinary than the forecasting models, but they have the advantage that they can make predictions even when the relevant circumstances change. In our case, since our goal is to make predictions after the situation changes due to the lifting of the lockdown measures, we need to consider a mechanistic model. However, it is widely known that the main limitation of mechanistic models is the difficulty of determining the parameters specifying such models. In this direction, a methodological advance was presented by the authors35, filling an important gap in the relevant literature: it was shown in35 that from the knowledge of the most reliable data of the epidemic in a given country, namely the cumulative number of deaths, it is possible to determine suitable combinations of the constant parameters (of the original model) which specify the differential equation characterizing the death dynamics. Furthermore, a robust numerical algorithm was presented for obtaining these parameters. One of these constants, denoted by c, is particularly important for the analysis of the effect of easing the lockdown conditions, because it is proportional to the number of contacts between asymptomatic individuals that are infected by SARS-CoV-2 and susceptible ones. Specifically, as the equations presented below will indicate, this coefficient is measured in units of inverse population (where the population represents the number of individuals to which we assign no units) times inverse days. This constant reflects the probability of infection given a contact which is proportional to the viral load (i.e., the viral concentration in the respiratory-tract fluid) of expelled respiratory droplets36. Easing the lockdown will lead to an increase of the value of this constant. Thus, in order to quantify this effect we assumed that the post-lockdown situation could be described by the same model but with c multiplied by an integer number (zeta), such as (zeta =2), or 3, etc. Assuming a fixed viral load emission (i.e., no face mask or similar protective measures), this would be tantamount to doubling or tripling the number of contacts per day. To put things into perspective, it is relevant to mention here that in the relevant literature a ballpark estimate for daily contacts of an individual is about 13.437.
We first applied the above algorithm to the case of the COVID-19 epidemic in Greece. However, the novelty and the main interest of the present work consists of the extension and application of the above methodology to two subpopulations. This situation is significantly more complicated than that of2 and is described by 12 ODEs involving 18 parameters (details are discussed in the “Methods” section). Using this extended formulation, we analysed the effect of easing the lockdown measures under two distinct possible scenarios: in the first, we examined what would happen if the interactions between older persons, namely persons above 40 years of age, as well as between older and younger persons, namely those below 40, continue to be dictated by the same restrictions as those of the lockdown period. However, we assumed that the interaction among the young was progressively more free. In the second case, we analysed the effect of easing the lockdown measures in the entire population without distinguishing the older from the young. In principle, the effect on deaths in the above two scenarios could be analyzed by the extension of the rigorous results of35. However, due to the sparsity of the deaths data (especially for the younger population), this approach is practically not possible at present. Thus, we supplemented the data for deaths for the two subpopulations with data for the cumulative numbers of reported infected.
Using four sets of data, namely the number of deaths and the number of reported infected for the older and the younger population we found that the above two alternatives would result in very different outcomes: in the first case, the total number of deaths of the two sub-populations and the number of total infections would be relatively small. In the second case, these numbers would be prohibitively high. Specifically, in the case of Greece, if the lockdown was to be continued indefinitely, our analysis suggests that the total numbers of deaths and infections would finally be around 165 and 2550, respectively. These numbers would remain essentially the same even if the lockdown measures for the interaction between the young people were eased substantially, provided that the interactions of older-older and older-young would remain the same as during the lockdown period. For example, even if the parameter measuring the effect of the lockdown restrictions on the young-young interactions were increased fourfold, the number of deaths and infections would be (according to the model extrapolation) 184 and 3585, respectively. On the other hand, even if the parameters characterizing all three interactions were increased only threefold, the relevant numbers would be 48144 and 1283462. It is clear that the latter numbers are prohibitive, suggesting that a generic release of the lockdown may be catastrophic.
In our view, the explanations provided in the “Methods” section for the assumptions of our model, which show that these assumptions are typical in the standard epidemiological models, substantiate the qualitative conclusions (and notes of caution) regarding the impact of the above two different types of exit policies. This may provide a sense of how a partial restoration of regular life activities can be achieved without catastrophic consequences, while the race for pharmacological or vaccine-based interventions that will lead to an end of the current pandemic is still ongoing. Importantly, we also offer some caveats emphasizing the qualitative nature of our conclusions and possible factors that may substantially affect the actual outcome of the lifting of lockdown measures.
Model setup: single population versus two age groups
We divide the population in two subpopulations, the young (y) and the older (o). In order to explain the basic assumptions of our model we first consider a single population, and then discuss the needed modifications in our case which involves two subpopulations. Let E(t) denote the exposed (but not infectious) population. An individual in this population, after a median 4-day period (required for incubation — see e.g.38) will either become sick or will be asymptomatic; an interval of 3-10 days captures 98% of the cases. The sick (infected) and asymptomatic populations will be denoted, respectively, by I(t) and A(t). The rate at which an exposed person becomes asymptomatic is denoted by a; this means that each day aE(t) persons leave the exposed population and enter the asymptomatic population. Similarly, each day sE(t) leave the exposed population and enter the sick population. These processes, as well as the subsequent movements are depicted in the flowchart of Fig. 1.
Figure 1

Flowchart of the populations considered in the model and the rates of transformation between them. The corresponding dynamical equations are Eqs. (1)–(6).

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The asymptomatic individuals recover with a rate (r_1), i.e., each day (r_1A(t)) leave the asymptomatic population and enter the recovered population, which is denoted by R(t). The sick individuals either recover with a rate (r_2) or they become hospitalized, H(t), with a rate h. In turn, the hospitalized patients also have two possible destinations; either they recover with a rate (r_3), or they become deceased, D(t), with a rate d.
It is straightforward to write the above statements in the language of mathematics; this gives rise to the equations (1)–(5) below:

$$begin{aligned} frac{dA}{dt}= a E – r_1 A end{aligned}$$
(1)

$$begin{aligned} frac{dI}{dt}= s E – (h + r_2) I end{aligned}$$
(2)

$$begin{aligned} frac{dH}{dt}= h I – (r_3+d) H end{aligned}$$
(3)

$$begin{aligned} frac{dR}{dt}= r_1 A + r_2 I + r_3 H end{aligned}$$
(4)

$$begin{aligned} frac{dD}{dt}= d H end{aligned}$$
(5)

$$begin{aligned} frac{dE}{dt}= c left[ T – (E+I+A+H+R+D)right] left( A + b Iright) – (a+s) E end{aligned}$$
(6)

It is noted that our model is inspired by various expanded versions of the classic SIR model adapted to the particularities of COVID-19 (such as the key role of the asymptomatically infected). It is, in particular, inspired by, yet not identical with that of14. In order to complete the system of equations (1)–(6), it is necessary to describe the mechanism via which a person can become infected. For this purpose we adopt the standard assumptions made in the typical epidemiological models, such as the SIR (susceptible, infected, recovered) model: let T denote the total population and let c characterize the number of contacts per day made by an individual with the capacity to infect (c is thought of as being normalized by T). Such a person belongs to I, A or H. However, for simplicity we assume that the hospitalized population cannot infect; this assumption is based on two considerations: first, the strict protective measures taken at the hospital, and second, the fact that hospitalized patients are infectious only for part of their stay in the hospital. The latter fact is a consequence of the relevant time scales of virus shedding in comparison to the time to hospitalization and the duration of hospital stay. The asymptomatic individuals are (more) free to interact with others, whereas the (self-isolating) sick persons are not. Thus, we use c to characterize the contacts of the asymptomatic persons and b to indicate the different infectiousness (due to reduced contacts/self-isolation) of the sick in comparison to the asymptomatic individuals.
The number of people available to be infected (i.e., the susceptible population) is (T-(E+I+A+H+R+D)). Indeed, the susceptible individuals consist of the total population minus all the individuals that are going or have gone through the course of some phase of infection, namely they either bear the infection at present ((E+A+I+H)) or have died from COVID-19 (D) or are assumed to have developed immunity to COVID-19 due to recovery (R). Hence, if we call the total initial individuals T, this susceptible population is given by the expression written earlier. The rate by which each day individuals enter E is given by the product of the above expression with (c(A+bI)). At the same time, as discussed earlier, every day ((a+s)E) persons leave the exposed population. It is relevant to note here that within this simpler model, it is possible to calculate the basic reproduction number (R_0), which is a quantity of substantial value in epidemiological studies32,33. In this model, this can be found to be33:

$$begin{aligned} R_0=frac{c T}{a+s}left[ frac{a}{r_1} + frac{b s}{r_2+h} right] . end{aligned}$$
(7)

This will be useful below for the purposes of finding the change in c (under lockdown) needed in order for transmission to cross the threshold of (R_0=1) and thus to lead to growth of the epidemic. In the particular case of the data shown in Table 1, (R_0=0.4084), in accordance with the lockdown situation associated with a controlled epidemic.
It is straightforward to modify the above model so that it can describe the dynamics of the older and younger subpopulations. Each subpopulation satisfies the same set of equations as those described above, except for the last equation which is modified as follows: the people available to be infected in each subpopulation are described by the expression given above where T, E, I, A, H, R, D have the superscripts (^o) or (^y), denoting older and young, respectively; (A+bI) is replaced in both cases by (A^o+A^y+b(I^o+I^y)) where for simplicity we have assumed that the infectiousness of the older and the young is the same. We have already considered the implications of the generalisation of the above model by allowing different parameters to describe the interaction of the older and young populations; this will be discussed in the “Methods” section. In what follows, we will discuss the results of this simpler “isotropic” interaction model.
Quantitative model findings
The parameters of the model are given in the flowchart of Fig. 1. Naturally, for the two-age model considered below, there is one set of such parameters associated with the younger population and one associated with the older one. The optimization routine used for the identification of these parameters is explained in detail in the “Methods” section. The parameters resulting from this optimization for the single population model are shown in Table 1, whereas for each of the two populations are given in Table 2. Clearly, many of these parameters are larger for the older population in comparison to the young, leading to a larger number of both infections and deaths in the older than in the young population.
Table 1 Optimized model parameters for the single population model, and the variation interval of each parameter within the optimization process (for further details, see “Methods” section).
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Table 2 Optimized (isotropic) model parameters for the young and older populations, and the variation interval of each parameter within the optimization process (for further details, see “Methods” section).
Full size table

Support for the validity of our model is presented in Fig. 2, which depicts its comparison (using the above optimized parameters) with the available data. The situation corresponding to keeping the lockdown conditions indefinitely, is the one illustrated in Fig. 2. In this case, the number of deaths and cumulative infections rapidly reaches a plateau, indicating the elimination of the infection. Here, we have optimized the model on the basis of data used from Greece39 between April 3rd and May 4th. It is noted that daily updates occurred at 3pm for the country of Greece, hence it is not clear up to what time the data are collected that are included in the daily report. We have assumed that the data reflect the infections and deaths present on that particular day. This possibly shifts the starting point of our count by a few hours, but should not change the overall result trends.
We next explain the implications of the model when different scenarios of ‘exit’ from the lockdown state are implemented. The relevant results are illustrated in Figs. 3, 4 and the essential conclusions are summarized in Table 3 for the numbers of deaths and cumulative infections, respectively. First, we need to explain the meaning of the parameter (zeta) appearing in the above tables: this parameter reflects the magnitude of the easing of the lockdown restrictions. Indeed, since the main effect of the lessening of these restrictions is that the number of contacts increases, we model the effect of easing the lockdown restrictions by multiplying the parameter c with a factor that we refer to as (zeta). The complete lockdown situation corresponds to (zeta)=1; the larger the value of (zeta), the lesser the restrictions imposed on the population. By employing the above quantitative measure of easing the lockdown restrictions, we consider in detail two distinct scenarios. In the first, which corresponds to the top rows of the Figures 3 and 4, we only allow the number of contacts of “young individuals with young individuals” (corresponding to the parameter (c^{yy}) mentioned in the “Methods” section) to be multiplied by the factor (zeta). This means that the lockdown measures are eased only with respect to the interaction of young individuals with other young individuals, while the interactions of the young individuals with the older ones, as well as the interactions among older individuals remain in the lockdown state. In the second scenario, corresponding to the bottom rows of the Figures 3 and 4, the restrictions of the lockdown are simultaneously eased in both the young and the older population; in this case all contacts are increased by the factor (zeta). It is noted that while we change c by this factor, we maintain the product cb at its previous value (i.e., we concurrently transform (crightarrow zeta c) and (brightarrow b/zeta)) considering that the sick still operate under self-isolation conditions and thus do not accordingly increase their number of contacts.
Table 3 Deaths D(t) and cumulative infections C(t) in the case of increasing of the number of contacts by (zeta). The second and fourth columns refer to the case for which the lockdown measures are eased for the young population, whereas the third and fifth column refer to the one where this occurs for both the young and older populations.
Full size table

Figure 2

Evolution of the current situation of deaths D(t) (left) and cumulative infections C(t) (right) in Greece, under the case of an indefinite continuation of the lockdown conditions. In this and all the figures that follow, the blue curve corresponds to the young population, while the red curve to the older population. The data for Greece from the 3rd of April to the 4th of May 2020 are depicted by dots. For the latter, alternate colors have been used (i.e., blue dots for the older population and red for the younger for clearer visualization).

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Fig. 3 corresponds to the case where the parameter (zeta) associated with the number of contacts between susceptible and asymptomatic individuals doubles. In this case, as also shown in Table 3, the situation does not worsen in a dramatic way. In particular, the number of deaths increases by 1, whereas the cumulative infections only increase by the small number of 58. In the second scenario where the number of contacts is doubled for both the young and the older populations, we find slightly larger (but not totally catastrophic) effects: the number of deceased individuals increases by 58 and the total number of infections grows by 1550.
Figure 3

Again the deaths D(t) and the cumulative infections C(t) are given for the case where the c factor (characterizing the number of contacts) amongst young individuals is doubled, but those of the older individuals (and of the young-older interaction) are kept fixed. This is shown in the top panels. In the bottom panels, the c’s of both young and old individuals are doubled.

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The situation becomes far more dire when the number of contacts is multiplied by a factor of 3 for both the young and older populations, meaning that the lockdown restrictions are eased significantly for the entire population. As shown in Table 3 and in Fig. 4, if the c’s of the young population only are multiplied by a factor of 3, then the deaths are increased by 3 and the infections by 198 (black line in the Figure and 3rd row of the Tables). This pales by comparison to the dramatic scenario when the c’s associated with both the young and older sub-populations are multiplied by 3; in this case, the number of deaths jumps dramatically to 48144, while the number of infections is a staggering 1283462, growing by about 500 times.
Figure 4

Same as reported in Fig. 3 but now where the contacts are multiplied by factors 3, 4 and 5. Full (dashed) lines hold for the young (older) population.

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An example corroborating the above qualitative trend can also be found in Fig. 4 and in the 4th and 5th rows of Table 3. Here, for e.g. (zeta =5), even the effect of releasing solely the young population leads to very substantial increases, namely to 6044 deaths and 306219 infections although of course it is nowhere near the scenarios of releasing both young and older populations. In the second scenario, the numbers are absolutely daunting: using the parameters of Table 2 we find that the number of deaths jumps to 83274 and the number of cumulative infections to 2221296.
Figure 5

Hospitalizations when only the young population (left) or both the young and older (right) population are released. Full (dashed) lines hold for the young (older) population.

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Finally, we show the prediction of the easing measures in the hospitalizations (i.e. daily occupied beds in hospitals). This is a crucial point to assess in order that the health system does not collapse because of COVID-19 patients. Figure 5 shows these trends for the above mentioned values of (zeta). In the case of releasing solely the young population (see left panel of the Figure), it is observed that the number of hospitalizations decreases monotonically except for (zeta =5), where the hospitalization peak is 523 for the young population and 1426 for the older one (values that are affordable by Greek health system); however, if both the young and older population are released (see right panel of the Figure), there is a monotonically decreasing behaviour only for (zeta =1) and 2. For higher (zeta) we observe that the height of the peak obviously increases with (zeta), while this peak also occurs earlier when the number of contacts is increased; for instance, for (zeta =3), the hospitalization peak number of the young population is 3844 whereas this value is 37030 for the older one, numbers that are, unfortunately, unaffordable for the Greek health system. These figures grow even further to 16869 and 163648 if (zeta =5).
In light of the above results, the significance of preserving the lockdown restrictions of the sensitive groups of the older population is naturally emerging. It can be seen that in the case where the number of contacts is roughly doubled, the behavior of release of young or young and older individuals is not dramatic (although even in this case releasing only the young population is, of course, preferable). Nevertheless, a more substantial release of the young population is still not catastrophic. On the other hand, the higher rates of infection, hospitalization and proneness to death of senior individuals may bring about highly undesirable consequences, should both the young and older members of the population be allowed to significantly increase (by 3 times or more) their number of contacts. More

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